Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,6,Mod(1,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 387.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(62.0685382676\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 43) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 17485 \nu^{7} + 269956 \nu^{6} + 1611305 \nu^{5} - 37491046 \nu^{4} + 6237658 \nu^{3} + \cdots - 6710235008 ) / 364843904 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 21039 \nu^{7} - 142704 \nu^{6} + 4791607 \nu^{5} + 20918022 \nu^{4} - 294848298 \nu^{3} + \cdots + 5711621760 ) / 182421952 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 22647 \nu^{7} + 254452 \nu^{6} + 3208555 \nu^{5} - 31393034 \nu^{4} - 140310162 \nu^{3} + \cdots - 1682303552 ) / 182421952 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12501 \nu^{7} + 88350 \nu^{6} + 1838311 \nu^{5} - 10354118 \nu^{4} - 64176154 \nu^{3} + \cdots - 471647712 ) / 91210976 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10033 \nu^{7} + 131102 \nu^{6} + 1204965 \nu^{5} - 17221020 \nu^{4} - 33518126 \nu^{3} + \cdots - 4059170624 ) / 45605488 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 124323 \nu^{7} + 267124 \nu^{6} + 18740383 \nu^{5} - 9550410 \nu^{4} - 725015914 \nu^{3} + \cdots + 11086446080 ) / 364843904 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} - 2\beta_{2} + 2\beta _1 + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 8\beta_{5} - 5\beta_{4} - \beta_{3} - 5\beta_{2} + 72\beta _1 + 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{7} + 89\beta_{6} + 28\beta_{5} - 81\beta_{4} - 23\beta_{3} - 259\beta_{2} + 302\beta _1 + 3124 \)
|
| \(\nu^{5}\) | \(=\) |
\( -195\beta_{7} + 243\beta_{6} + 1092\beta_{5} - 603\beta_{4} - 71\beta_{3} - 1119\beta_{2} + 6762\beta _1 + 9282 \)
|
| \(\nu^{6}\) | \(=\) |
\( 163 \beta_{7} + 8597 \beta_{6} + 5444 \beta_{5} - 7077 \beta_{4} - 3425 \beta_{3} - 29617 \beta_{2} + \cdots + 285638 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 26531 \beta_{7} + 37567 \beta_{6} + 127500 \beta_{5} - 65855 \beta_{4} - 10463 \beta_{3} + \cdots + 1337942 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−8.95911 | 0 | 48.2657 | 61.2927 | 0 | −166.001 | −145.726 | 0 | −549.129 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.21373 | 0 | −4.81697 | 9.80186 | 0 | −11.1041 | 191.954 | 0 | −51.1043 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.65705 | 0 | −18.6260 | 107.102 | 0 | −25.5214 | 185.142 | 0 | −391.677 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.582753 | 0 | −31.6604 | −27.7074 | 0 | −103.690 | 37.0983 | 0 | 16.1466 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.08717 | 0 | −15.2950 | 61.4284 | 0 | −184.774 | −193.303 | 0 | 251.069 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.06235 | 0 | 17.8767 | −73.4416 | 0 | −4.24720 | −99.7434 | 0 | −518.670 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.09504 | 0 | 33.5297 | 63.3756 | 0 | 223.489 | 12.3830 | 0 | 513.028 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.1681 | 0 | 92.7262 | 10.1483 | 0 | 135.849 | 678.196 | 0 | 113.337 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(43\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 387.6.a.c | 8 | |
| 3.b | odd | 2 | 1 | 43.6.a.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 688.6.a.e | 8 | ||
| 15.d | odd | 2 | 1 | 1075.6.a.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.6.a.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 387.6.a.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 688.6.a.e | 8 | 12.b | even | 2 | 1 | ||
| 1075.6.a.a | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 12T_{2}^{7} - 117T_{2}^{6} + 1502T_{2}^{5} + 3358T_{2}^{4} - 49104T_{2}^{3} - 39464T_{2}^{2} + 439936T_{2} + 259776 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(387))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 12 T^{7} + \cdots + 259776 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 5172924974752 \)
|
| $7$ |
\( T^{8} + \cdots + 116222354316288 \)
|
| $11$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $13$ |
\( T^{8} + \cdots - 20\!\cdots\!44 \)
|
| $17$ |
\( T^{8} + \cdots + 37\!\cdots\!33 \)
|
| $19$ |
\( T^{8} + \cdots + 20\!\cdots\!68 \)
|
| $23$ |
\( T^{8} + \cdots - 18\!\cdots\!83 \)
|
| $29$ |
\( T^{8} + \cdots - 39\!\cdots\!36 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!13 \)
|
| $37$ |
\( T^{8} + \cdots - 10\!\cdots\!04 \)
|
| $41$ |
\( T^{8} + \cdots + 17\!\cdots\!57 \)
|
| $43$ |
\( (T + 1849)^{8} \)
|
| $47$ |
\( T^{8} + \cdots - 19\!\cdots\!04 \)
|
| $53$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots - 68\!\cdots\!68 \)
|
| $61$ |
\( T^{8} + \cdots + 23\!\cdots\!84 \)
|
| $67$ |
\( T^{8} + \cdots + 69\!\cdots\!48 \)
|
| $71$ |
\( T^{8} + \cdots - 15\!\cdots\!64 \)
|
| $73$ |
\( T^{8} + \cdots + 80\!\cdots\!92 \)
|
| $79$ |
\( T^{8} + \cdots + 19\!\cdots\!12 \)
|
| $83$ |
\( T^{8} + \cdots - 19\!\cdots\!08 \)
|
| $89$ |
\( T^{8} + \cdots - 13\!\cdots\!64 \)
|
| $97$ |
\( T^{8} + \cdots + 38\!\cdots\!17 \)
|
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